Problem: The geometric sequence $a_i$ is defined by the formula: $a_1 = 8$ $a_i = a_{i - 1} \cdot (-1.5)$ Find the sum of the first $20$ terms in the sequence. Choose 1 answer: Choose 1 answer: (Choice A) A $-2.47\cdot10^{17}$ (Choice B) B $-53{,}220.11$ (Choice C) C $-17{,}734.70$ (Choice D) D $ -10{,}637.62 $
Solution: Getting started Let's write out the first few terms of the series: $8 -12 + 18...$ We're dealing with a geometric series because each term is multiplied by $-1.5$ to get the next term. We need a formula to compute the sum of the terms. Formula for geometric series The sum $S_n$ of a finite geometric series is $S_n = \dfrac{a_1(1-r^n)}{1-r}$ where $a_1$ is the first term, $r$ is the common ratio, and $n$ is the number of terms. What do we need to use the formula? The first term $(a_1 = {8})$ and the number of terms $(n = {20})$ are given in the question. The common ratio $r$ is ${-1.5}$ because each term is multiplied by ${-1.5}$ to get the next term. Find the sum $(S_n)$ of the series $\begin{aligned} S_n &= \dfrac{a_1(1-r^n)}{1-r} \\\\ S_{{20}}&=\dfrac{{8}(1-\left({-1.5}\right)^{{20}})}{1-\left({-1.5}\right)} \\\\ S_{{{20}}} &\approx -10{,}637.62 \end{aligned}$ The answer $ -10{,}637.62 $